Algebra & Surds Test B

0 of 29 parts answered

Test

Time allowed: 80 minutesTotal: 68 marks12 questions

Candidate: Hana

Instructions

Answer all questions.
Write your answer to each part on the answer line. Use the maths keys that appear at the bottom of the screen for √, powers and other symbols.
Show all your working. You can earn marks for correct method even if your final answer is wrong.
You must not use a calculator.
Give answers in exact form — fractions or surds — unless the question says otherwise.
The number of marks for each question or part question is shown in brackets [ ].

Question 1

(a)

Solve the simultaneous equations. Show all your working.

4 x + 3 y = 25

2 x - 3 y = - 1

x =

y =

[3]

Question 2

(a)

Solve the simultaneous equations. Show all your working.

3 x + 4 y = 5

5 x - 3 y = 18

x =

y =

[4]

Question 3

2 pens and 5 notebooks cost $15.50. 4 pens and 3 notebooks cost $13.50.

(a)

Write down two equations, using

p

for the cost of a pen and

n

for the cost of a notebook, in dollars.

[2]

(b)

Solve your equations to find the cost of one pen and the cost of one notebook.

Pen $

Notebook $

[3]

Question 4

(a)

Factorise

x^{2} + 2 x - 35

[2]

(b)

Hence solve

x^{2} + 2 x - 35 = 0

x =

x =

[1]

(c)

Solve

3 x^{2} - 10 x - 8 = 0

by factorising.

x =

x =

[3]

Question 5

(a)

Write

x^{2} + 8 x + 3

in the form

(x + p)^{2} + q

p =

q =

[2]

(b)

Write

x^{2} - 5 x + 1

in the form

(x + p)^{2} + q

p =

q =

[2]

Question 6

(a)

Write

3 x^{2} + 18 x + 4

in the form

a (x + b)^{2} + c

a =

b =

c =

[3]

(b)

Write

2 x^{2} + 10 x - 1

in the form

a (x + b)^{2} + c

a =

b =

c =

[3]

Question 7

(a)

By completing the square, solve

x^{2} - 4 x - 8 = 0

. Give your answers in the form

a \pm b 3

x =

x =

[3]

(b)

By completing the square, solve

2 x^{2} + 4 x - 3 = 0

. Leave your answers in surd form.

x =

x =

[4]

Question 8

(a)

Find the value of the discriminant of

3 x^{2} - 4 x + 2 = 0

[2]

(b)

Hence write down the number of real roots of

3 x^{2} - 4 x + 2 = 0

[1]

(c)

Without solving the equation, show that

9 x^{2} + 6 x + 1 = 0

has one repeated root.

[2]

(d)

Use the quadratic formula to solve

x^{2} + 5 x + 2 = 0

. Give your answers in the form

\frac{a \pm b}{c}

x =

x =

[3]

Question 9

(a)

The equation

x^{2} - k x + 16 = 0

has one repeated root. Find the two possible values of

k

k =

k =

[3]

(b)

The equation

x^{2} + 6 x + k = 0

has no real roots. Find the range of values of

k

[2]

Question 10

(a)

Simplify

45

[1]

(b)

Simplify

98

[1]

(c)

Write

200 - 32

in the form

k 2

[2]

(d)

Simplify

32 \times 410

, giving your answer in the form

k 5

[2]

Question 11

(a)

Expand and simplify

(5 + 2) (3 - 2)

[2]

(b)

Expand and simplify

(7 - 3) (7 + 3)

[2]

(c)

Expand

(3 + 5)^{2}

, giving your answer in the form

a + b 5

a =

b =

[2]

Question 12

(a)

Rationalise the denominator and simplify

\frac{10}{5}

[2]

(b)

Rationalise the denominator and simplify

\frac{6}{2 + 3}

[3]

(c)

Write

\frac{2 + 3}{4 - 3}

in the form

\frac{a + b 3}{c}

, where

a

b

and

c

are integers.

[3]

End of paper

Mark scheme

Algebra & Surds Test B · 68 marks

Question	Answer	Marks	Partial marks
1(a)	$x = 4$ , $y = 3$	3	M1 for correctly eliminating one variable, e.g. $6 x = 24$ A1 for $x = 4$ A1 for $y = 3$
2(a)	$x = 3$ , $y = - 1$	4	M1 for multiplying one or both equations so that the coefficients of $x$ (or $y$ ) are equal, e.g. $9 x + 12 y = 15$ and $20 x - 12 y = 72$ M1 for correctly adding (or subtracting) to eliminate a variable, e.g. $29 x = 87$ A1 for $x = 3$ A1 for $y = - 1$
3(a)	$2 p + 5 n = 15.5$ and $4 p + 3 n = 13.5$	2	B1 for each correct equation
3(b)	Pen $1.50, notebook $2.50	3	M1 for a correct method to eliminate one variable (FT their equations) A1 for $p = 1.5$ A1 for $n = 2.5$
4(a)	$(x + 7) (x - 5)$	2	M1 for $(x + a) (x + b)$ where $ab = - 35$ or $a + b = 2$
4(b)	$- 7$ and $5$ FT their (a)	1
4(c)	$- \frac{2}{3}$ and $4$	3	M2 for $(3 x + 2) (x - 4)$ or M1 for $(3 x + a) (x + b)$ where $ab = - 8$ or $a + 3 b = - 10$
5(a)	$(x + 4)^{2} - 13$	2	B1 for $p = 4$ B1 for $q = - 13$
5(b)	$(x - \frac{5}{2})^{2} - \frac{21}{4}$ oe	2	B1 for $p = - \frac{5}{2}$ B1 for $q = - \frac{21}{4}$
6(a)	$3 (x + 3)^{2} - 23$	3	B1 for $a = 3$ B1 for $b = 3$ B1 for $c = - 23$
6(b)	$2 (x + \frac{5}{2})^{2} - \frac{27}{2}$ oe	3	B1 for $a = 2$ B1 for $b = \frac{5}{2}$ B1 for $c = - \frac{27}{2}$
7(a)	$2 + 23$ and $2 - 23$	3	M1 for $(x - 2)^{2} - 4 - 8$ oe M1 for $(x - 2)^{2} = 12$ or $x - 2 = \pm 12$ A1 for both answers — accept $2 \pm 12$ for M2 only
7(b)	$\frac{- 2 + 10}{2}$ and $\frac{- 2 - 10}{2}$ oe	4	M1 for $x^{2} + 2 x - \frac{3}{2} = 0$ or $2 (x^{2} + 2 x) - 3$ M1 for $(x + 1)^{2} - 1$ seen M1 for $(x + 1)^{2} = \frac{5}{2}$ oe A1 for both answers, e.g. $- 1 \pm \frac{5}{2}$
8(a)	$- 8$	2	M1 for $(- 4)^{2} - 4 \times 3 \times 2$
8(b)	0 (no real roots) FT their (a)	1
8(c)	$b^{2} - 4 a c = 6^{2} - 4 \times 9 \times 1 = 36 - 36 = 0$ , so one repeated root	2	M1 for $6^{2} - 4 \times 9 \times 1$ A1 for $= 0$ with the conclusion
8(d)	$\frac{- 5 + 17}{2}$ and $\frac{- 5 - 17}{2}$	3	M1 for $\frac{- 5 \pm 5 ^{2} - 4 \times 1 \times 2}{2 \times 1}$ B1 for 17 seen as the discriminant
9(a)	$8$ and $- 8$	3	M1 for $(- k)^{2} - 4 \times 1 \times 16 = 0$ M1 for $k^{2} = 64$ or B1 for one correct value only
9(b)	$k > 9$	2	M1 for $6^{2} - 4 \times 1 \times k < 0$ or B1 for $k = 9$ or $k \geq 9$
10(a)	$35$	1
10(b)	$72$	1
10(c)	$62$	2	M1 for $102$ or $42$ seen
10(d)	$245$	2	M1 for $1220$ or $20 = 25$ seen
11(a)	$13 - 22$	2	M1 for three of $15$ , $- 52$ , $+ 32$ , $- 2$ correct
11(b)	46	2	M1 for $49 + 73 - 73 - 3$ or $49 - 3$
11(c)	$14 + 65$	2	M1 for $9 + 35 + 35 + 5$ oe
12(a)	$25$	2	M1 for $\frac{10 5}{5}$
12(b)	$12 - 63$	3	M1 for multiplying by $\frac{2 - 3}{2 - 3}$ B1 for denominator $4 - 3 = 1$
12(c)	$\frac{11 + 6 3}{13}$	3	M1 for multiplying by $\frac{4 + 3}{4 + 3}$ M1 for numerator $8 + 23 + 43 + 3$ or denominator 13

M method mark · A accuracy mark (depends on the M mark) · B mark for a correct result or statement, independent of method · oe or equivalent · cao correct answer only · FT follow through after an earlier error · isw ignore subsequent working

29 parts not answered yet.